Eventual gonality conjecture for king's graphs

Let $K_{m\times n}$ denote the king's graph on an $m\times n$ grid, with $4\leq m\leq n$. The gonality of a graph is denoted by $\operatorname{gon}(G)$. Eventual gonality conjecture. For $4\leq m\leq n$, we have

$$\operatorname{gon}(K_{m\times n})=3m-2.$$

This conjecture is motivated by computational evidence that the gonality is already $10$ for $K_{4\times4}$ and $K_{4\times5}$, although the stated formula remains to be established for the indicated family of king's graphs.